By Deo S.

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9: Choosing a Hermitian inner product on E , the endomor- phisms of E × S 1 form a vector space End(E × S 1 ) with a norm α = sup|v|=1 |α(v)| . The triangle inequality holds for this norm, so balls in End(E × S 1 ) are convex. The subspace Aut(E × S 1 ) of automorphisms is open in the topology defined by this norm since it is the preimage of (0, ∞) under the continuous map End(E × S 1 )→[0, ∞) , inf (x,z)∈X × S 1 | det(α(x, z))| . Thus to prove the first statement of the lemma it α will suffice to show that Laurent polynomials are dense in End(E × S 1 ) , since a sufficiently close Laurent polynomial approximation to f will then be homotopic to f via the linear homotopy t + (1 − t)f through clutching functions.

2. We use the notation ‘ H ’ for the canonical line bundle over CP1 since its unit sphere bundle is the Hopf bundle S 3 →S 2 . To determine the ring structure in K(S 2 ) we have only to express the element H 2 , represented by the tensor product H ⊗ H , as a linear combination of 1 and H . The claim is that the bundle (H ⊗ H) ⊕ 1 is isomorphic to H ⊕ H . 2, these are the clutching functions f g ⊕ 11 and f ⊕ g where both f and g are the function z (z) . As we showed there, the clutching functions f g⊕ 11 and f ⊕g are always homotopic, so this gives the desired isomorphism (H ⊗ H) ⊕ 1 ≈ H ⊕ H .

The left square can be rewritten − − → − − → X ∧ SA → − − − X ∧ ( X ∪ CA) SA →−−−− X ∪ CA where the horizontal maps collapse the copy of X in X ∪CA to a point, the left vertical map sends (a, s) ∈ SA to (a, a, s) ∈ X ∧ SA , and the right vertical map sends x ∈ X to (x, x) ∈ X ∪ CA and (a, s) ∈ CA to (a, a, s) ∈ X ∧ CA . Commutativity is then obvious. It is often convenient to have an unreduced version of the groups K n (X) , and this can easily be done by the simple device of defining K n (X) to be K n (X+ ) where X+ is X with a disjoint basepoint labeled ‘+’ adjoined.